Imaginary and Complex Numbers

Question 1

If z = a + bj, what is the complex conjugate of z?

Answer 1

If z = a + jb

then the complex conjugate is:

ź (or z*) = a - jb

Question 2

What is j¹?

Answer 2

j¹ = j

or

j = ‚/-1

Question 3

What does j² evaluate to?

Answer 3

j² = -1

Question 4

What does j³ evaluate to?

Answer 4

j³ = -j

Question 5

What does j⁴ evaluate to?

Answer 5

j⁴ = j²•j²

= -1 • -1

= 1

Question 6

What does √-1 x √-1 evaluate to?

Answer 6

√-1 x √-1 = j x j

= j²

= -1

Question 7

What is the argument of a complex number?

give the formula

Answer 7

The argument of a complex number is the angle between the Re axis and the position vector of the complex number as depicted on an Argand diagram, such that -π < arg z ≤ π. This holds true as long as z ≠ 0.

Arg (z) = tan^-1 Im(z) / Re(z), z ≠ 0

also

Arg (z) = sin^-1 Im(z) / |z|, z ≠ 0

also

Arg (z) = cos^-1 Re(z) / |z|, z ≠ 0

Question 8

What is the modulus of an imaginary number?

Give formula

Answer 8

The Modulus of an imaginary number is the scalar length (magnitude) of the position vector of the imaginary number as depicted on an Argand diagram.

|z| = r = Sqrt (Re[z]² + Im[z]²)

Question 9

Write the complex number z = x + jy in polar form.

Answer 9

Given that |z| = r = Sqrt (Re[z]² + Im[z]²)

and therefore Re(z) = r cosø

and also Im(z) = r sinø

z = r cosø + j r sinø

= r (cosø + j sinø)

Question 10

Convert the Complex number z = r (cosø + j sinø) to Exponential form.

Answer 10

Given that (cosø + j sinø) = e^jø

z = r (cosø + j sinø)

becomes

z = r(e^jø)

=re^jø

if and only if ø is in radians

Question 11

Simplify j⁴²

Answer 11

j⁴² = ( j⁴)¹⁰ ( j²)

= (1)¹⁰ (-1)

= -1

Question 12

Simplify j¹²

Answer 12

j¹² = ( j⁴)³

= (1)³

= 1

Question 13

Simplify j¹¹

Answer 13

j¹¹ = ( j⁴)² ( j³)

= (1)² ( - j)

= - j

Question 14

Simplify j ^{- 8}

Answer 14

j ^{- 8} = ( j ^{- 4})²

= (1)²

= 1

Question 15

Simplify j^-1

Answer 15

Start with j² = -1

divide both sides by j

j = -1/j

= - j^-1

multiply both sides by -1

j^-1 = - j

Question 16

Simplify j^-2

Answer 16

j^-2 = (j²)^-1

= (-1)^-1

= -1

Question 17

Simplify j ^-3

Answer 17

j ^-3 = (j ^-2) j ^-1

= (-1) (-j)

= j

Question 18

SImplify j ^-4

Answer 18

j ^-4 = (j ^-2)²

= (-1)²

= 1

Question 19

Simplify j ^{- 19}

Answer 19

j ^{- 19} = ( j^-4)⁴ ( j ^{- 3})

= (1)⁴ ( j)

= j

Question 20

Simplify j ^{- 30}

Answer 20

j ^{- 30} = ( j ^{- 4})⁷ ( j ^{- 2})

= (1) (-1)

= -1

Question 21

Simplify j ^{- 15}

Answer 21

j ^{- 15} = ( j ^{- 4})³ ( j ^{- 3})

= (1)³ ( j)

= j

Question 22

Simplify j ^{- 32}

Answer 22

j ^{- 32} = ( j ^{- 4})⁸

= (1)⁸

= 1

Question 23

Simplify j ^{- 13}

Answer 23

j ^{- 13} = ( j ^{- 4})³ ( j ^-1)

= (1)³ (- j)

= - j

Question 24

Simplify j ^{- 23}

Answer 24

j ^{- 23} = ( j ^{- 4})⁵ ( j ^{- 3})

= (1)⁵ ( j)

= j

Question 25

In the complex number **z = 3 + j5** what is the real part and what is the imaginary part? Use proper notation.

Answer 25

The parts of the complex number **z = 3 + j5** are as follows: **Re(z) = 3** (the real part) **Im(z) = 5** (the imaginary part) So, a complex number **z = Re(z) + j Im(z)**

Question 26

Complete the following complex number addition: **(4 + j5) + (3 - j2)**

Answer 26

## Footnote **(4 + j5) + (3 - j2) = 4 + j5 + 3 - j2** **= (4 + 3) + j(5 - 2)** **= 7 + j3**

Question 27

Complete the following complex number subtraction: **(4 + j7) - (2 - j5)**

Answer 27

## Footnote **(4 + j7) - (2 - j5) = 4 + j7 - 2 + j5** **= (4 - 2) + j(7 + 5)** **= 2 + j12**

Question 28

Write down the general example of complex number addition.

Answer 28

## Footnote **z₁ + z₂ = (a + jb) + (c + jd)** **= a + jb + c + jd** **= (a + c) + j(b + d)**

Question 29

Complete the following complex number addition: **(5 + j7) + (3 - j4) - (6 - j3)**

Answer 29

## Footnote **(5 + j7) + (3 - j4) - (6 - j3) = 5 +j7 + 3 - j4 - 6 + j3** **= (5 + 3 - 6) + j(7 - 4 + 3)** **= 2 + j6**

Question 30

Write down the general example of complex number multiplication.

Answer 30

## Footnote **z₁ z₂ = (a + jb)(c + jd)** **= ac + jad + jbc + j²bd** **= ac + jad + jbc - bd** **= (ac - bd) + j(ad +bc)**

Question 31

Multiply the following complex numbers: **z₁ = 3 + j4** **z₂ = 2 + j5**

Answer 31

## Footnote **z₁ z₂ = (3 + j4)(2 + j5)** **= 6 + j15 + j8 + j²20** **= 6 + j23 - 20** **= -14 + j23**

Question 32

Multiply the following complex numbers: **z₁ = (3 + j4)** **z₂ = (2 - j5)** **z₃ = (1 - j2)**

Answer 32

## Footnote **z₁ z₂ z₃ = (3 + j4)(2 - j5)(1 - j2)** **= (3 + j4)(2 - j4 - j5 + j²10)** **= (3 + j4)(2 - j9 - 10)** **= (3 + j4)(-8 - j9)** **= -24 - j27 - j32 - j²36** **= -24 - j59 + 36** **= 12 - j59**

Question 33

Multiply the complex numbers **(5 + j8)(5 - j8)** Note: complex conjugates

Answer 33

## Footnote **(5 + j8)(5 - j8) = 25 + j40 - j40 - j²64** **= 25 - ( -64)** **= 25 + 64** **= 89**

Question 34

Under what circumstances is the complex number **z = a + jb** purely real?

Answer 34

The complex number **z = a + jb** is purely real when **b = 0**

Question 35

Under what circumstances is the complex number **z = a + jb** purely imaginary?

Answer 35

The complex number **z = a + jb** is purely imaginary when **a = 0**

Question 36

How do you divide one Complex number (**z₁ = a + jb**) by another (**z₂ = c + jd**)?

Answer 36

**z₁/z₂ = (a + jb) / (c + jd)** to simplify we can make the denominator entirely real using the conjugate complex number **z₂\*** **z₁z₂\* / z₂z₂\* = (a + jb)(c - jd) / (c + jd)(c - jd)** **= ((ac + bd) + j(bc - ad)) / c² + d²**

Question 37

What is an **Argand Diagram**?

Answer 37

An **Argand Diagram** is a way of visually representing **complex numbers**. In an **Argand Diagram** the complex number is plotted on a pair of co-ordinate axes, where the horizontal axis represents **Real numbers** and the vertical axis represents **Imaginary numbers.**

Question 38

Given the **modulus (r)** and the **argument (ø)** how can you work out **Re(z)** and **Im(z)**?

Answer 38

Given **r** and **ø** we can determine **Re(z)** as follows: **Re(z) = rcos(ø)** Similarly, we can determine **Im(z)** as follows: **Im(z) = rsin(ø)**

Question 39

How would you multiply the following complex numbers: **z₁ = r (cos a + j sin a)** **z₂ = s (cos b + j sin b)** hint: use exponential form

Answer 39

**z₁ = r (cos a + j sin a)** **= re^ja** **z₂ = s (cos b + j sin b)** **= se^jb** **z₁z₂ = re^ja • se^jb** **= rs(e^ja • e^jb)** **= rs(e^ja+jb)** **=rse^j(a+b)** Note: rse^j(a+b) = rs(cos(a+b) + jsin(a+b))

Question 40

What does the **polar/****exponential form**tell us about**complex number multiplication**? hint: magnitudes and angles

Answer 40

The **polar form** tells us that when we **multiply complex numbers** we: - **multiply** the **magnitudes** and - **add** the **angles** e. g. **r (cos(a) + jsin(a)) • s (cos(b) + jsin(b))** **= rs (cos(a + b) + jsin(a +b))**

Question 41

Divide the complex number **z₁ = 7(cos (7π/6) + j sin (7π/6))** by **z₂ = cos (7π/4) + j sin (7π/6)** hint: exp

Answer 41

**z₁ = 7(cos (7π/6) + j sin (7π/6))** **= 7e^j(7π/6)** and **z₂ = cos (7π/4) + j sin (7π/6)** **= e^j(7π/4)** therefore **z₁ / z₂ = 7e^j(7π/6) / e^j(7π/4)** **= 7e^j(7π/6) • e^{- j(7π/4)}** **= 7e^{j(7π/6 - 7π/4)}** **= 7e^{- j7π/12}** **z₃ = 7(cos (7π/12) - jsin (7π/12))**

Question 42

How would you divide the following complex numbers: **z₁ = r (cos a + j sin a)** by **z₂ = s (cos b + j sin b)** hint: use exponential form

Answer 42

**z₁ = r (cos a + j sin a)** **= re^ja** and **z₂ = s (cos b + j sin b)** **= se^jb** therefore **z₁ / z₂ = re^ja /** **se^jb** **= (r/s)e^ja • e^{- jb}** **= (r/s)e^{j(a - b)}** **= r/s (cos (a-b) + jsin (a-b))**

Question 43

What does the **polar/exponential** form tell us about **complex number division**? hint: **magnitudes** and **angles**

Answer 43

The **polar form** tells us that when we **divide complex numbers** we: - **divide** the **magnitudes** and - **subtract** the **angles** e. g. **r (cos(a) + jsin(a)) / s (cos(b) + jsin(b))** **= r/s (cos(a - b) + jsin(a -b))**

Question 44

Does (z₁z₂)\* = z₁\*z₂\* ?

Answer 44

Yes it does. let **z₁ = a + jb** and **z₂ = c + jd** **z₁z₂** = **(a + jb)(c + jd)** **= (ac - bd) + j(ad + bc)** Therefore: **(z₁z₂)\*** = **(ac - bd) - j(ad + bc)** Given that **z₁\* = a - jb** and **z₂\* = c - jd** it can be seen that: **z₁\*z₂\* = (a - jb)(c - jd)** **= (ac - bd) + j(-ad -bc)** **= (ac - bd) - j(ad + bc)** Thus: **(z₁z₂)\* = z₁\*z₂\***

Question 45

What can we say about **zz\***? hint: it something to do with **mod** **z**

Answer 45

Given that: **z = a + jb** and **z\* = a - jb** we can see that: **zz\*** = **(a + jb)(a - jb)** **= a² + b²** Given that **|z|** = **√(a² + b²)** it can be seen that: **zz\* = |z|²**